针对非线性退化抛物方程，提出一种采用中心多项式和基于 L² 范数光滑度指示子的 Z 型权重的六阶中心 WENO 格式，避免负线性权重并提升分辨率。
摘要：
本文针对非线性退化抛物方程，发展了一种具有 Z 型非线性权重的新型六阶有限差分中心加权本质无振荡（WENO）格式。为避免负线性权重，在 WENO 重构中引入了中心多项式。我们基于 L² 范数光滑度指示子选择 Z 型非线性权重，使新的 WENO 格式具有更精确的分辨率。文中还证实，所提出的带有所设计非线性权重的中心 WENO 格式在光滑区域达到六阶精度。一维和二维数值算例展示了该中心 WENO 格式改进后的性能。
英文摘要：
In this paper we develop a new sixth-order finite difference central weighted essentially non-oscillatory (WENO) scheme with Z-type nonlinear weights for nonlinear degenerate parabolic equations. The centered polynomial is introduced for the WENO reconstruction in order to avoid the negative linear weights. We choose the Z-type nonlinear weights based on the $L^2$-norm smoothness indicators, yielding the new WENO scheme with more accurate resolution. It is also confirmed that the proposed central WENO scheme with the devised nonlinear weights achieves sixth order accuracy in smooth regions. One- and two-dimensional numerical examples are presented to demonstrate the improved performance of the proposed central WENO scheme.
A High-Order Modified Finite Volume WENO Method on 3D Cartesian Grids
School of Mathematics and Systems Science, Beihang University（北京航空航天大学数学与系统科学学院）；ICMSEC and LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
The aim of this study is to develop a novel WENO scheme that improves the performance of the well-known fifth-order WENO methods. The approximation space consists of exponential polynomials with a tension parameter that may be optimized to fit the the specific feature of the data, yielding better results compared to the polynomial approximation space. However, finding an optimal tension parameter is a very important and difficult problem, indeed a topic of active research. In this regard, this study introduces a practical approach to determine an optimal tension parameter by taking into accoun…
Sixth-order parabolic equation on an interval: Eigenfunction expansion, Green's function, and intermediate asymptotics for a finite thin film with elastic resistance
Exp-ParaDiag: Time-Parallel Exponential Integrators for Parabolic PDEs
Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram（印度科学教育与研究学院特里凡得琅数学学院）
Implicit and implicit--explicit high-order BDF methods for coupled elliptic--parabolic systems
University of Ioannina（约阿尼纳大学）；Institute of Applied and Computational Mathematics, FORTH（FORTH应用与计算数学研究所）；Lanzhou University（兰州大学）；Wuhan University（武汉大学）
The recently proposed high-order TENO scheme [Fu et al., Journal of Computational Physics, 305, pp.333-359] has shown great potential in predicting complex fluids owing to the novel weighting strategy, which ensures the high-order accuracy, the low numerical dissipation, and the sharp shock-capturing capability. However, the applications are still restricted to simple geometries with Cartesian or curvilinear meshes. In this work, a new class of high-order shock-capturing TENO schemes for unstructured meshes are proposed. Similar to the standard TENO schemes and some variants of WENO schemes, t…
In this paper an even higher-order compact GKS up to sixth order of accuracy will be constructed for the shock and acoustic wave computation on unstructured mesh. The compactness is defined by the physical domain of dependence for an unstructured triangular cell, which may involve the closest neighbors of neighboring cells. The compactness and high-order accuracy of the scheme are coming from the consistency between the high-order initial reconstruction and the high-order gas evolution model under GKS framework. The high-order evolution solution at a cell interface provides not only a time-acc…
We develop a general framework for construction and analysis of discrete extension operators with application to unfitted finite element approximation of partial differential equations. In unfitted methods so called cut elements intersected by the boundary occur and these elements must in general by stabilized in some way. Discrete extension operators provides such a stabilization by modification of the finite element space close to the boundary. More precisely, the finite element space is extended from the stable interior elements over the boundary in a stable way which also guarantees optima…
High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids
University of Trento（特伦托大学）；Université de Pau et des Pays de l’Adour（波城及阿杜尔地区大学）；Sobolev Institute of Mathematics（索博列夫数学研究所）；Novosibirsk State University（新西伯利亚国立大学）
Energy Stable and Efficient Finite-Difference Nonlinear Multigrid Schemes for the Modified Phase Field Crystal Equation
North Carolina State University（北卡罗来纳州立大学）；University of Massachusetts, North Dartmouth（麻省大学诺斯敦分校）；The University of Tennessee, Knoxville（田纳西大学诺克斯维尔分校）
